Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Suppose $a$ and $b$ are elements of a finite field of order $2^n$ with $n$ odd and $a^2+ab+b^2=0$. Is it necessary that both $a$ and $b$ must be zero ?

I understand that the field has characteristic $2$ but don't know how to use the fact that $n$ is odd, please help.

share|improve this question
7  
If $b$ were not 0, say, then the ratio $a/b$ is a root of $x^2 + x + 1$ (divide the equation by $b^2$). What does that tell you about the subfield ${\mathbf F}_2(a/b)$ inside your finite field? –  KCd Dec 22 '12 at 14:41
2  
Well, if $n$ is $2$ for example you can take $b=1$ and $a$ to be either of the elements that are neither $1$ nor $0$. –  Chris Eagle Dec 22 '12 at 14:41
    
@Chris: It was specified that $n$ is odd. –  Cameron Buie Dec 22 '12 at 14:46
    
@CameronBuie: Yes, I know. The OP seemed to be unsure why this was relevant. –  Chris Eagle Dec 22 '12 at 14:47
1  
KCd nailed it. Another equivalent way to think about this is to determine the fields $GF(2^n)$ that contain a primitive third root of unity. –  Jyrki Lahtonen Dec 22 '12 at 18:34
show 2 more comments

1 Answer 1

up vote 4 down vote accepted

If $b$ were not 0 then $a/b$ would be a root of $x^2 + x + 1$, which is irreducible over ${\mathbf F}_2$. Look at the size of the field ${\mathbf F}_2(a/b)$ and the size of the field you are working in that has order $2^n$.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.